If f(x) = (4x + 3)/ (6x – 4), x ≠ (2/3) show that fof(x) = x, for all x ≠ (2/3). What is the inverse of f?

Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ y

Express R and R^-1 as sets of ordered pairs. Determine also

(i) the domain of R^‑1

(ii) The Range of R.

A function f: R → R is defined as f(x) = x³ + 4. Is it a bijection or not? In case it is a bijection, find f⁻¹ (3).

If f: R → R be defined by f(x) = x³ −3, then prove that f⁻¹ exists and find a formula for f⁻¹. Hence, find f⁻¹ (24) and f⁻¹ (5).

Consider f: R⁺ → [−5, ∞) given by f(x) = 9x² + 6x − 5. Show that f is invertible with f^-1(x) = \( \dfrac{\sqrt{(x+6)-1}}{3}\)

Consider f: R → R⁺ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with inverse f⁻¹ of f given by f⁻¹(x) = \( \sqrt{ (x-4)}\) where R⁺ is the set of all non-negative real numbers.